Blow-up solutions on a sphere for the 3d quintic NLS in the energy space

Abstract

We prove that if $u(t)$ is a log-log blow-up solution, of the type studied by Merle-Raphaël (2001-2005), to the $L^2$ critical focusing NLS equation $i\partial_t u +Δu + |u|^{4/d} u=0$ with initial data $u_0\in H^1(\mathbb{R}^d)$ in the cases $d=1, 2$, then $u(t)$ remains bounded in $H^1$ away from the blow-up point. This is obtained without assuming that the initial data $u_0$ has any regularity beyond $H^1(\mathbb{R}^d)$. As an application of the $d=1$ result, we construct an open subset of initial data in the radial energy space $H^1_{rad}(\mathbb{R}^3)$ with corresponding solutions that blow-up on a sphere at positive radius for the 3d quintic ($\dot H^1$-critical) focusing NLS equation $i\partial_tu + Δu + |u|^4u=0$. This improves Raphaël-Szeftel (2009), where an open subset in $H^3_{rad}(\mathbb{R}^3)$ is obtained. The method of proof can be summarized as follows: on the whole space, high frequencies above the blow-up scale are controlled by the bilinear Strichartz estimates. On the other hand, outside the blow-up core, low frequencies are controlled by finite speed of propagation.